Question

A silver concentration cell is set up at \(25^{\circ} \mathrm{C}\) as shown below:The \(\mathrm{AgCl}(s)\) is in excess in the left compartment. a. Label the anode and cathode, and describe the direction of the electron flow. b. Determine the value of \(K_{\text {sp }}\) for \(\mathrm{AgCl}\) at \(25^{\circ} \mathrm{C}\).

Short answer

In the silver concentration cell, the anode is where the less concentrated solution of Ag+ ions oxidizes, and the cathode is where the more concentrated solution of Ag+ ions reduces. The electron flow is from the anode to the cathode. Using the Nernst equation and the given temperature of 25°C, we find that the solubility product constant (K_sp) for AgCl is 1.

Step-by-step solution

Step 1:

Identify the anode, cathode, and direction of electron flow In a silver concentration cell, the half-reactions are as follows: Anode (oxidation): \(Ag_{(s)} \rightarrow Ag^+_{(aq)} + e^-\) Cathode (reduction): \( Ag^+_{(aq)} + e^- \rightarrow Ag_{(s)} \) The more concentrated solution of Ag+ ions will reduce at the cathode, while the less concentrated solution will oxidize at the anode. The electron flow will be from the anode to the cathode.

Step 2:

Determine the cell potential. From the Nernst equation, the cell potential is given by: \(E_{cell} = E^\circ - \dfrac{RT}{nF} \cdot \ln(Q)\) Where: - \(E_{cell}\) is the cell potential - \(E^\circ\) is the standard cell potential, which is given as 0 for a concentration cell - R is the ideal gas constant, which is 8.314 J/mol·K - T is the temperature in Kelvin, which is 298.15 K - n is the number of moles of electrons transferred during the reaction (n = 1 for the given reaction) - F is the Faraday's constant, which is 96485 C/mol - Q is the reaction quotient, given by \(\dfrac{[Ag^+]_{cathode}}{[Ag^+]_{anode}}\) Since the standard cell potential is 0, the Nernst equation simplifies to: \(E_{cell} = - \dfrac{RT}{nF} \cdot \ln(Q)\)

Step 3:

Determine the value of the solubility product constant (K_sp) for AgCl. The solubility product constant (K_sp) for the dissolution of AgCl is given by: \(K_{sp} = [Ag^+][Cl^-]\) Since the reaction is at equilibrium, the cell potential is also 0. Therefore, the Nernst equation becomes: \(0 = - \dfrac{RT}{nF} \cdot \ln(Q)\) From this equation, we can determine Q: \(\ln(Q) = 0\) \(Q = 1\) Since the reaction quotient is equal to the solubility product constant (K_sp) at equilibrium, we have: \(K_{sp} = \dfrac{[Ag^+]_{cathode}}{[Ag^+]_{anode}} = 1\) Thus, the value of K_sp for AgCl at 25°C is 1.

Key concepts

Electrochemistry

Electrochemistry is a branch of chemistry that studies chemical reactions which take place at the interface of an electrode, usually a solid metal or a semiconductor, and an ionic conductor, the electrolyte. These reactions involve electron transfer between the electrode and the electrolyte or species in solution. If a chemical reaction is driven by an externally applied voltage, as in electrolysis, or if it gives rise to a voltage, as in a battery, it is called an electrochemical reaction.

In the case of the silver concentration cell from the exercise, we have two compartments containing electrolytic solutions connected by a salt bridge, with electrodes dipped into them. Here, the key electrochemical process is the movement of silver ions (Ag⁺) from one electrode to the other while releasing or absorbing electrons in the process.

Nernst Equation

The Nernst equation is a fundamental equation in electrochemistry that relates the reduction potential of a half-cell to the standard electrode potential, temperature, activity, and reaction quotient of the redox reaction. It can be expressed as:
\[E = E^\circ - \dfrac{RT}{nF} \ln(Q)\]
Where:

• \(E\) is the cell potential,
• \(E^\circ\) is the standard cell potential,
• R is the gas constant (8.314 J/mol·K),
• T is the absolute temperature in Kelvin,
• F is Faraday's constant (96,485 C/mol),
• n is the number of moles of electrons exchanged in the half-cell reaction,
• Q is the reaction quotient.

The Nernst equation is pivotal in our understanding of the potential of the silver concentration cell, as it accounts for the effects of ion concentration on the cell potential.

Solubility Product Constant

The solubility product constant, commonly abbreviated as K_sp, is a value used to describe the equilibrium between a solid and its respective ions in a solution. For any ionic compound like AgCl (Silver chloride), the product of the molar concentrations of the dissolved ions (Ag⁺ and Cl^- in this case), each raised to the power of its coefficient in the balanced equation, gives the solubility product constant:
\[K_{{sp}} = [Ag^+][Cl^-]\]
This equation shows that the solubility product is constant at a given temperature and is characteristic of the specific ionic compound. The solubility product is related to the compound’s solubility; the smaller the product, the less soluble the compound. Understanding K_sp is crucial in determining the concentration of ions in a saturated solution and predicting the formation of a precipitate.

Cell Potential

Cell potential, also referred to as electromotive force (EMF) of a cell, is the measure of the potential difference between two half-cells in an electrochemical cell. It is a thermodynamic quantity that indicates how much electrical energy is obtainable from a redox reaction. The cell potential can be determined using the Nernst equation and depends on the standard cell potential, temperature, and the concentration of the reactants and products involved in the cell reactions.

For the silver concentration cell, the cell potential is the driving force for the transfer of silver ions. Since it is a concentration cell with identical electrodes, the standard cell potential is zero, and the cell potential is determined only by the ion concentration difference between the two compartments.

Chemical Equilibrium

Chemical equilibrium is the state of a reversible reaction where the rate of the forward reaction equals the rate of the backward reaction, and the concentrations of reactants and products remain unchanged over time. At equilibrium, the reaction has reached a state of balance and the chemical composition of the system does not undergo any further change.

For the silver concentration cell, equilibrium comes into play in terms of the solubility of AgCl. As the Ag⁺ ions are converted to solid Ag at the cathode and solid Ag is converted to Ag⁺ ions at the anode, the system reaches a chemical equilibrium when the solid AgCl stops dissolving. At this point, the reaction quotient (Q) equals K_sp, and, according to the Nernst equation, the cell potential at equilibrium will be zero, indicating there is no driving force for the reaction to proceed in either direction.